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330 U. Bunke, T. Schick, M. Spitzweck, and A. Thom6.2.8 The construction E 7! T .E/ refines to a functorT W Prin B .G/ ! Tors.G jB /from the category of G-principal bundles Prin B .G/ over B to the category of G jB -torsors Tors.G jB / over B.Lemma 6.13. The functoris an equivalence of categories.T W Prin B .G/ ! Tors.G jB /Proof. It is a consequence of the Yoneda Lemma that T is an isomorphism on the levelof morphism sets. It remains to show that the underlying sheaf T of a G jB -torsor isrepresentable by a G-principal bundle. Since T is locally isomorphic to G jB this istrue locally. The local representing objects can be glued to a global representing object.We can now prolong the chain of bijections (52) toŒB; BG Š H 0 .Prin B .G// Š H 0 .Tors.G jB //Š H 0 .EXT.Z jB ;G jB // Š Ext 1 Sh Ab S=B .Z jB ;G jB / Š H 1 .BI G/:(53)6.2.9 Let S be some site and F 2 S be a sheaf of abelian groups. By Gerbe.F /we denote the two-category of gerbes with band F over S. It is well-known thatisomorphism classes of gerbes with band F are classified by Ext 2 Sh Ab S .ZI F/, i.e. thereis a natural bijectiond W H 0 .Gerbe.F // ! Ext 2 Sh Ab S .ZI F/:6.2.10 Let H ! G be a homomorphism of topological abelian groups with kernelK WD ker.H ! G/. Our main example is R n ! T n with kernel Z n .Definition 6.14. Let T be a space. An H -reduction of a G-principal bundle E D.E ! B/ on T is a diagram.F; /W F E; T Bwhere F ! T is an H -principal bundle, and is H -equivariant, where H acts onE via H ! G. An isomorphism .F; / ! .F 0 ; 0 / of H -reductions over T is anisomorphism f W F ! F 0 of H -principal bundles such that 0 ı f D . Let R E H .T /denote the category of H -reductions of E.